Utilizing Bayesian inference in accelerated testing models under constant stress via ordered ranked set sampling and hybrid censoring with practical validation

This research investigates the application of the ordered ranked set sampling (ORSSA) procedure in constant-stress partially accelerated life-testing (CSPALTE). The study adopts the assumption that the lifespan of a specific item under operational stress follows a half-logistic probability distribution. Through Bayesian estimation methods, it concentrates on estimating the parameters, utilizing both asymmetric loss function and symmetric loss function. Estimations are conducted using ORSSAs and simple random samples, incorporating hybrid censoring of type-I. Real-world data sets are utilized to offer practical context and validate the theoretical discoveries, providing concrete insights into the research findings. Furthermore, a rigorous simulation study, supported by precise numerical calculations, is meticulously conducted to gauge the Bayesian estimation performance across the two distinct sampling methodologies. This research ultimately sheds light on the efficacy of Bayesian estimation techniques under varying sampling strategies, contributing to the broader understanding of reliability analysis in CSPALTE scenarios.


Type-I hybrid censoring
In the realm of reliability studies and survival analysis, type-I hybrid censoring emerges as a strategic approach that amalgamates the principles of both type-I and type-II censoring.Let's dissect these concepts to gain a comprehensive understanding.
Type-I censorship involves terminating an experiment at a pre-defined time, without considering whether the event of interest (such as failure) has occurred or not.Conversely, type-II censoring concludes the investigation after a specific number of incidents (failures) have been observed.Individuals or groups that are still ongoing at the conclusion of the study, having not experienced the event, are deemed censored.Now, in the landscape of type-I hybrid censoring, both these censoring processes are intricately intertwined.In a type-I hybrid scenario, the study concludes either when a predetermined time limit is reached or when a specified number of events have transpired.This dual criterion for concluding the study is denoted as T ⋆ = min(T s:m , τ ) , where T s:m represents the time when the specified count s of failures occurs, and τ signifies the predetermined time limit.
In practical terms, when subjecting a batch of m units to testing, the individual lifetimes of these units are considered as independent and identically distributed random variables, denoted by T 1 , T 2 , . . ., T m .The testing process dynamically concludes based on the occurrence of either a specified count of failures or the elapse of a predefined time.This nuanced methodology allows investigators to optimize study time, striking a balance between the benefits of type-I and type-II censorship, offering flexibility and efficiency in reliability analysis.
Hybrid censoring has gained increased prominence, prompting numerous researchers to explore statistical analysis techniques for different distributions within the realm of hybrid censoring in reliability studies.Noteworthy contributors to this area include [21][22][23][24][25] , among other scholars.

Ranked set sampling
The primary method for gathering data through sampling is known as SRSA, which involves randomly selecting sampling units.Assigning ranks to various sampling units without the need for actual measurements can be relatively straightforward and cost-effective in various fields (e.g., fisheries and medical research).This is especially useful when it would be expensive and time-consuming to measure the variable of interest directly.To obtain more representative samples from the wider population in such circumstances, sampling designs that emphasise ranks can be used, which will ultimately increase the efficacy of statistical analysis.Ranked set sampling (RSSA) was originally suggested by McIntyre 26 , and multiple studies, whether through numerical or theoretical means, have illustrated the advantages of employing RSSA-based statistical methods compared to their counterparts in the SRSA approach.RSSA is a unique and efficient sampling technique that aims to enhance the accuracy of statistical estimates while reducing the cost and effort of data collection.Unlike conventional random sampling methods, where individual items are randomly selected from a population, RSSA involves selecting entire sets of items and arranging them in order based on a specific criterion.Additional details about the RSSA technique can be explored in references such as [27][28][29][30] . (1) Within the realm of science, RSSA finds numerous applications, particularly within environmental and ecological studies, where the primary emphasis lies in the development of cost-effective and efficient sampling methods.Reference 26 played a pioneering role in establishing the theoretical framework of RSSA, especially in situations where the quantification of sample items proves overly expensive or unfeasible.In such cases, although the variable to be monitored can be ranked more easily and inexpensively than directly measured, as argued by the authors.The authors asserted that when it comes to estimating the mean of a population, utilizing RSSA holds greater value and is more advantageous compared to employing SRSA.This notion was further supported by the mathematical demonstration presented in Ref. 31 , indicating that the mean estimation through RSSA surpasses the performance of SRSA.Within the realm of reliability analysis, Ref. 32 delved into the task of estimating the reliability of stress-strength, denoted as P(Y * < X * < Z * ) , applicable to a unit characterized by a strength denoted as X * , along with lower bound stress denoted as Y * , and upper bound stress denoted as Z * using RSSA technique.
Reference 33 detailed the Bayesian approach for parameter estimation in PSALTE data, assuming an exponential distribution.The researchers in 34 studied different point predictors, including the best-unbiased, conditional median, and Bayes point predictors.Their study focused on predicting future order statistics based on PSALTE data that are believed to follow a Rayleigh distribution.
This paper introduces the implementation of CSPALTE through ORSSA.The research is carried out with the underlying assumption that the lifespan of an item under operational stress follows a half-logistic distribution.Utilizing type-I hybrid censoring, Bayesian estimation is investigated, with a particular focus on estimating pertinent parameters.Both SLFU and ASLFU are employed in the investigation.Generally, we are motivated to present this paper for the following reasons: 1. Traditional CSALTE methods often assume complete and precise information about the lifetimes of products, leading to potential biases and inaccurate reliability predictions.The incorporation of ordered ranked set sampling facilitates more precise and efficient parameter estimation by strategically selecting the most informative subsets, thereby reducing estimation errors and enhancing the reliability of the results.2. The proposed methodology integrates type-I hybrid censoring, accommodating scenarios where the continuous monitoring of product lifetimes is impractical or costly.This feature enables the optimization of resources by combining continuous monitoring with periodic inspections, ensuring a balance between data accuracy and cost-effectiveness in the context of reliability assessments.3. Real-life reliability (engineering) data often exhibit skewness and non-normality, violating assumptions of classical statistical methods.Bayesian estimation, coupled with ordered ranked set sampling, provides a robust framework capable of handling skewed and non-normally distributed data.This allows for more accurate modeling of the underlying reliability distributions, particularly in situations where traditional methods may falter.4. The proposed methodology is designed to be versatile and applicable across diverse industries where ALTE is crucial for ensuring the longevity and reliability of products.Industries such as electronics, automotive, and aerospace can benefit from this research by obtaining more accurate estimates of product lifetimes, ultimately leading to improved decision-making processes and enhanced product quality.5.This work contributes to the advancement of Bayesian statistical techniques in the realm of reliability engineering.By combining ORSSA with Bayesian estimation under type-I hybrid censoring, we aim to push the boundaries of current methodologies, opening avenues for further exploration and refinement of Bayesian approaches in reliability analysis.6. Applying the Bayesian estimation to provide a natural way to quantify and propagate uncertainty through the analysis, which is essential in reliability engineering where uncertainty is inherent.7.That being said, the proposed Bayesian estimation framework for CSPALTE based on ORSSA under type-I hybrid censoring addresses key challenges in reliability engineering, offering a methodological leap towards more accurate, resource-efficient, and robust parameter estimation in the assessment of product lifetimes.
The following portions of the paper are arranged as follows: Under the CSPALTE, the ORSSA is described in section "Model description".In sections "Bayes estimation based on ORSSA under CSPALTE" and "Bayes estimation based on a SRSA under CSPALTE", respectively, Bayesian estimation under type-I hybrid censoring using ORSSA and SRSA is investigated.Section "Real data set: Light-Emitting Diodes" provides a real-world example.Sections "Simulation study" and "Conclusions, discussion, and some potential future points" contain the simulation studies and findings, respectively.

Model description
In the realm of reliability testing, the CSPALTE method, as detailed in 14 and 35 , adheres strictly to either utilization or accelerated conditions for every tested item.A pivotal player in this methodology is the tampered RV (TRV) model, extensively explained in 36 .This model finds its application within the CSPALTE technique.The crux of CSPALTE lies in the segregation of total test items into two distinct groups.The first group undergoes testing under standard stress denoted as v 0 , while the second group is subjected to accelerated stress denoted as v 1 .It's a meticulous approach, strategically dividing the items to gather insights under different stress conditions.
As per the CSPALTE technique, let's delve into the specifics.Consider m 1 as the count of items from group 1 that fail under stress v 0 with their lifetime RV denoted as Z = T .The method operates under the assump- tion that the CDFU and PDFU of an item's lifetime Z in group 1 are defined by ( 1) and ( 2), respectively (with a simple substitution of t by z).This nuanced approach provides a comprehensive understanding of the lifetime characteristics of the tested items, paving the way for robust reliability analysis.
Furthermore, consider the quantity m 2 as the count of items within group 2 that undergo failure when exposed to stress v 1 .The TRV model, a key player in this analysis, dictates that the lifetime Z of an item experiencing stress v 1 can be derived by dividing its lifetime under standard stress v 0 by an acceleration factor µ greater than 1 (expressed as Z = T/µ ).This relationship highlights the role of acceleration in influencing the lifespan of the tested items.
Expanding upon this concept, the CSPALTE technique leverages the TRV model to unravel the statistical characteristics of items in group 2. Drawing from the established CDFU under utilization conditions, referred to as CDFU (1), and the PDFU under utilization conditions, referred to as PDFU (2) for the lifetime variable T, we can now articulate the subsequent expressions for the CDFU and PDFU of an item's lifetime variable Z in group 2. These expressions serve as a vital lens through which we can dissect and understand the distribution patterns of lifetimes for items undergoing accelerated stress conditions, offering a nuanced perspective in the realm of reliability analysis, where CDFU (1) (after replacing t by z) and CDFU (3) can be merged in a single equation as follows: For Similarly, for PDFUs (2) and ( 4),

RSSA under CSPALTE
Under CSPALTE, the following appro ach can be considered to obtain a RSSA of size 2m: 1. Assign a fixed value for m. 2. Begin by selecting a total of 2m 2 items from a specific population, and then meticulously organize them into 2m SRSAs.Ensuring uniformity, each of these SRSAs is designed to have an identical size, denoted by m, see Fig. 1.This systematic approach to sampling guarantees a comprehensive representation of the population, as the selected items are distributed evenly across the multiple SRSAs.The uniform size of each sample aids in maintaining statistical integrity, facilitating a thorough and unbiased exploration of the chosen subset.This strategic division sets the stage for a meticulous analysis of the population's characteristics, allowing for meaningful insights to be gleaned from the sampled items.3. Set r = 1. 4. In the systematic implementation of CSPALTE, the initial step involves a meticulous division of the items earmarked for examination into two distinct groups.Each of these groups is meticulously constructed to represent a SRSA, comprising precisely m items.This strategic sampling ensures a representative subset of the population under scrutiny, setting the stage for a comprehensive analysis of the items' performance under varying stress conditions.Within the CSPALTE framework, a pivotal facet of the experimental design involves subjecting these two groups of items to distinct stress environments.The first group under- .This standard stress level serves as a control or baseline condition against which subsequent observations and results can be compared.It provides a reference point for gauging the inherent reliability and longevity of the items when subjected to a stress level commonly encountered in their intended operational environment.Simultaneously, the second group undergoes a more rigorous examination under an accelerated stress denoted as v 1 .This accelerated stress level is deliberately chosen to exert higher demands on the items, simulating conditions that expedite the aging or wear-and-tear processes.By subjecting this group to an elevated stress environment, the CSPALTE methodology aims to accelerate the occurrence of failures and gather valuable data on the items' response to heightened operational stresses.5. Order the SRSAs in each group without practical measurement.6.In the O-th ordered SRSA, O = 1, 2 , a single item is measured.7.In group O, the r-th smallest item, say Z O,rr , O = 1, 2 , is measured.8. Set r = r + 1 .If r = m + 1 , then halt the previous steps and proceed to Step 10.If not, the smallest item in group O, say Z O,r+1r+1 , O = 1, 2 , is measured.9. Iterate Steps 4-8.10.A RSSA of size 2m is now generated under CSPALTE, see Fig. 1. 11.The one-cycle RSSA of size 2m not only streamlines the data collection process but also enhances the statistical efficiency of the estimation procedures, ensuring that the information gleaned from the experiment is both robust and insightful.To provide a clearer understanding, let's delve into the specifics of the notation.Consider the notation Z 2,44 , where it signifies the fourth smallest item within the fourth sample belonging to the second group.This notation is instrumental in precisely identifying and referencing specific elements within the structured sampling framework.In essence, it efficiently communicates the sample index, group affiliation, and the position of the item within that particular sample.12. Iterate Steps 2-9 U cycles to obtain a RSSA of size 2mU .The obtained data are shown by Now, suppose that Z RSSA is a 1-cycle RSSA from a population under CSPALTE with CDFU (5) and PDFU (6).Then, the CDFU and PDFU of Z O,kk , i = 1, 2 , indicated by H O,k:m and h O,k:m , are in fact the CDFU and PDFU of the k-th order statistic in group i.So, they can be expressed as follows, see 37 and 38 , where H O (z) and h O (z) are given by ( 5) and (6), respectively.CDFU (7) and PDFU (8) can be rewritten as where

RSSA from CSPALTE under type-I hybrid censoring
In the context of CSPALTE, the acquisition of the type-I hybrid censored ordered 1-cycle RSSA unfolds through the following procedure: For each O in the range of 1-2, assume that the experimental time τ O and the count of observed failures s are predetermined before the commencement of the experiment.Let it be known that the lifetimes of m units designated for testing (where m exceeds another pre-assigned value s) follow independent RVs with non-identically distributed (IRVNID) CDFU H O,k:m (z) [see Eq. ( 9)] and PDFU h O,k:m (z) [see Eq. (10)].
The experimenter then makes a decisive determination to conclude the experiment at either the occurrence of the s-th failure or at time τ O , depending on which event transpires first.
If the s-th failure transpires prior to the designated time τ O , the protocol dictates the removal of all the remain- ing surviving units, denoted as m − s , from the test.This action effectively brings the test to a conclusion at the point of the s-th failure time.
Conversely, if the s-th failure does not manifest before the specified time τ O and only b O failures, where b O is less than or equal to s, occur before τ O , then, at the precise moment τ O , the remaining surviving units, totaling m − b O , are removed from the test.This outcome leads to the termination of the test at τ O .Now, suppose that the observations given in Step 11, Section "RSSA under CSPALTE", have been ordered such that y O,j ≡ z O,jj:mm .Then type-I hybrid censoring gives rise to the following two observational cases: Following this process, the resultant data set derived from the described operation is termed as a type-I hybrid 1-cycle ORSSA.It is specifically designated by: where

Bayes estimation based on ORSSA under CSPALTE
Based on the insights provided by Balakrishnan 39 and considering the 1-cycle ORSSA (12), the formulation of the likelihood function (LHF) under the conditions of type-I hybrid censoring can be articulated as follows:  (9) and PDFU (10) in (13), the LHF can take the next form Based on the CDFU (5) and the PDFU (6) and using the next relations (12) Vol  (17), the LHF can be expressed as where where c O ,m γ O ,ϕ O is given by (19).For U-cycle ORSSA, the LHF under type-I hybrid censoring is then given by Using the relations given in (17), the LHF can be rewritten as where (17) 2,c O ,m

Prior and posterior distributions
In Bayesian inference, it is assumed that the unknown parameters are considered as RVs controlled by a collective prior density function.With the use of past data and learned information, this prior density function can be determined.When prior knowledge is limited, Bayesian inference can be applied with non-informative priors.
In this context, we consider η and µ are independent variables with the following joint prior density, π 1 (η) has Weibull model with two parameters b 1 and b 2 as follows: while π 2 (µ) is non-informative prior of the following form: Using ( 32) and ( 33), joint prior density (31) becomes From ( 27) and ( 34), the joint density function of the posterior model of η and µ may be expressed as: where

Bayesian estimation under asymmetric and symmetric loss functions
Bayesian estimation involves a statistical approach where uncertainty is captured by using probability distributions.The choice of a certain loss-function (LFU) plays a crucial role in this process.The LFU quantifies the discrepancy between predicted values and actual outcomes.In the context of Bayesian estimation, both SLFU and ASLFU are utilized.SLFUs treat overestimation and underestimation equally.They aim to minimize the overall error between predictions and true values, regardless of the direction of the error.( 28) 2,c ρ,O ,m Vol.:(0123456789) On the other hand, ASLFUs assign different penalties to overestimation and underestimation.This reflects situations where the cost of making one type of error is higher than the other.Asymmetric losses are especially useful when dealing with skewed or heavy-tailed data.
The selection of a suitable LFU depends on the specific problem and the associated costs of prediction errors.Bayesian estimation allows incorporating prior information, uncertainty, and the chosen LFU to make more informed and context-aware decisions.
The superiority of estimators for model parameters is evident when employing ASLFUs compared to estimators obtained using SLFUs.Numerous authors have explored the Bayesian estimation of underlying parameters using both SLFU and ASLFU, as evidenced in various works, including references such as 40 -43 .
The Bayes estimates (BESs) of η and µ , due to SLFU and ASLFU are 1.The squared error LFU (SELFU) is defined as follows: where ˆ is the estimation of the parameter .The BESs of , based on the SELFU, is given by From ( 35) and ( 36) the BESs of η and µ , based on the SELFU, are then given, respectively, by 2. The weighted SELFU (WSELFU) is defined as follows: The BES of , based on the WSELFU, is given by From ( 35) and ( 39) the BESs of η and µ , based on the WSELFU, are then given, respectively, by 3. The precautionary LFU (PRLFU) is defined as follows: .
Vol:.( 1234567890) The BES of , based on the LGLFU, is given by From ( 35) and ( 45) the BESs of η and µ , based on the LGLFU, are then given, respectively, by 5.The linear exponential LFU (LINEXLFU) is defined as follows: where ζ = � − � .The BES of , based on the LINEXLFU, is given by From ( 35) and (48) the BESs of η and µ , based on the LINEXLFU, are then given, respectively, by The BES of , based on the GELFU, is given by From ( 35) and (51) the BESs of η and µ , based on the GELFU, are then given, respectively, by

Real data set:Light-emitting diodes
Light-emitting diodes (LEDs) find widespread application in various semiconductor devices, particularly in television screens and color displays.These LEDs are composed of thinly layered semiconductor materials, featuring significant doping.The specific spectral wavelength emitted during forward-biasing of the LED is contingent upon the semiconductor material and the extent of doping.In this study, adopting the methodology delineated by Dey et al. 44 , we explore occurrences of failures both under normal operating conditions (sample size: 58) and accelerated stress conditions (sample size: 58) after 1000 hours, as summarized in Table 1.
Before delving further, we employ the statistic test of Kolmogorov-Smirnov (K-SM) along with the p-value for each stress level.The objective is to evaluate the appropriateness of fitting the data with the half-logistic distribution and its CDFU [see Eq. ( 5)].The outcomes, presented in Table 2, affirm that CDFU (5) aptly captures the actual data at each stress level, given that all p-values surpass 0.05.This statistical fit is visually reinforced by plotting the empirical CDFU of the real data against the CDFU of the half-logistic distribution [see Eq. ( 5)], as depicted in Fig. 2.
We choose hyperparameter values of b 1 = 1.15 , and b 2 = 0.966 to yield a population parameter value of η = 0.9197 using (32).www.nature.com/scientificreports/ In the context of CSPALTE, we opt for five SRSAs, each comprising ten elements.These SRSAs are subsequently partitioned into two groups, each containing five elements, as delineated in the second and fourth columns of Table 3. Within each SRSA, the first and second sets of elements are selected from the data pertaining to standard and accelerated stresses, respectively.The RSSA technique is then applied to these SRSAs, resulting in the 1-cycle and 2-cycle RSSAs, which is presented in the third and fifth columns of Table 3.For both the 1-cycle and 2-cycle scenarios, we employ the type-I hybrid censoring procedure on the data originating from the SRSA (specifically, the first sample) and the associated RSSAs, all of which are detailed in Table 3.

Remark 1
In cases where the hyperparameters are not known, one approach is to employ the empirical Bayes method to estimate them based on historical samples 45 .Another option is to utilize the hierarchical Bayes method, where an appropriate prior for the hyperparameters is employed 46 .Also, the elicitation method can be considered an approach for determining the values of hyperparameters within a prior distribution for one or more parameters of the sampling distribution, see 47 and the references therein.

Simulation study
In this section, we embark on a comprehensive exploration within the CSPALTE framework.Our focus is on a rigorous comparative Monte Carlo simulation study that involves an in-depth examination of the ORSSA and SRSA techniques, specifically under the conditions of type-I hybrid censoring.Furthermore, we delve into the computation and comparison of BESs for both η and µ , employing both SLFU and ASLFU.The orchestrated Monte Carlo simulation procedure unfolds as follows: Vol:.( 1234567890

Conclusions, discussion, and some potential future points
In recent years, the ORSSA technique has garnered significant attention, primarily due to its efficiency in producing estimates compared to simple random sampling methods.This paper delves into the application of the ORSSA within the context of CSPALTEs when the lifetime of a unit exposed to use stress adheres to the half-logistic distribution.The study adopts type-I hybrid censoring and explores both SLFU and ASLFU to examine the Bayesian estimations of the parameters under consideration, comparing the performance of ORSSA and SRSA.
To substantiate the theoretical findings, actual datasets have been employed, offering practical insight into the presented results.Additionally, a comprehensive simulation study, complemented by numerical computations, Additionally, the paper explores an innovative approach to Bayesian estimation in the context of CSPALTEs by combining ORSSA and type-I hybrid censoring.The study provides a valuable contribution to the field of reliability engineering and statistical methodology for ALTE, offering a robust framework for modeling and estimating the lifetime distribution of a product under different stress levels.
One of the significant highlights of this research is the integration of ORSSA and type-I hybrid censoring.The ORSSA is a non-traditional sampling technique that has gained attention due to its efficiency in reducing sampling costs while maintaining the accuracy of parameter estimates.By applying ordered ranked set sampling in combination with type-I hybrid censoring, the study harnesses the strengths of both methods, leading to improved estimation precision and cost-effectiveness.This integration is a novel approach and addresses the practical constraints often faced in ALTE scenarios.The use of Bayesian estimation techniques in this study is noteworthy.Bayesian methods offer a flexible and coherent framework for handling uncertainty and making inferences about the parameters of interest.The incorporation of prior information and the posterior distribution of parameters in the analysis allows for more robust and informative estimates.Bayesian estimation is especially advantageous when dealing with limited data, as is often the case in ALTE.
The proposed methodology has practical applications in various industries, including aerospace, automotive, electronics, and more, where reliability and product lifespan are critical factors.By improving the accuracy and efficiency of parameter estimation in CSALTEs experiments, this research provides a valuable tool for engineers and researchers seeking to optimize product design and assess the reliability of components and systems.
While this study successfully integrates ORSSA and type-I hybrid censoring for Bayesian estimation in CSPALTs, there are several avenues for future research.Further exploration of different censoring strategies, the impact of prior distributions on estimation accuracy, and the development of computational tools for practical implementation could enhance the applicability of the proposed methodology.
Below, we provide some relevant potential future point: 1. Investigate the applicability of Bayesian estimation in the context of other hybrid censoring schemes beyond type-I.Explore how the proposed methodology can be adapted or extended to accommodate different censoring structures, such as type-II or generalized hybrid censoring, to enhance its versatility and generalizability.2. Explore the incorporation of dynamic stress levels in CSPALTEs.Investigate Bayesian estimation techniques that can effectively handle scenarios where stress levels change over time, allowing for a more realistic representation of the dynamic nature of stress conditions in practical applications.3. Assess the robustness of the Bayesian estimation approach to various model assumptions.Investigate the impact of deviations from assumed distributional forms for the lifetime data and explore methods to enhance the methodology's resilience to model misspecifications.4. Further delve into the incorporation of informative prior distributions.Explore how prior knowledge about the system or product under study can be effectively integrated into the Bayesian framework, providing more informed and accurate estimates of the parameters of interest.5. Conduct comparative studies with alternative estimation approaches in the context of CSPALTs.Compare the performance of the Bayesian methodology with frequentist counterparts or other Bayesian methods to identify strengths, limitations, and areas for improvement.6. Extend the simulation studies to investigate the performance of the proposed Bayesian estimation method under conditions of small sample sizes.Assess the accuracy and reliability of parameter estimates when data availability is limited, which is often the case in real-world applications.7. Explore the practical implementation and efficacy of the proposed Bayesian estimation approach in various industries.Consider applications in fields such as automotive, aerospace, electronics, and healthcare to validate the method's utility and performance across diverse domains.8. Consider the development of user-friendly software tools that implement the Bayesian estimation methodology for CSPALTEs.Facilitate broader adoption and application of the proposed approach by providing accessible computational tools for practitioners and researchers.
Figure 2. Comparative representation of histograms and empirical CDFUs against PDFUs and CDFUs of the half-logistic distribution, employing the provided data fromTable 1 across two different stress levels.

Table 4 .
The real SRSAs (specifically, the first sample for each stress) under type-I hybrid censoring.

Table 5 .
The real ORSSAs under type-I hybrid censoring.

Table 8 .
BESs of η and µ based on 1000 SRSAs and ORSSAs in a 1-cycle.hasbeen executed to assess the efficacy of Bayesian estimations derived from ORSSA in contrast to SRSA.The numerical outcomes affirm the superiority of ordered ranked set sampling as a sampling technique, reinforcing its growing prominence in research and applications.

Table 9 .
BESs of η and µ based on 1000 SRSAs and ORSSAs in a 2-cycle.Investigate Bayesian estimation techniques that can accommodate scenarios involving multiple failure modes.Extend the methodology to handle complex systems where different failure mechanisms contribute to the overall reliability, providing a more comprehensive analysis.10.Explore the integration of Bayesian estimation with reliability growth models.Investigate how the proposed methodology can be synergistically employed with models that capture the improvement in reliability over time, offering insights into the system's evolving performance.

Table 10 .
BESs of η and µ based on 1000 SRSAs and ORSSAs in a 3-cycle.